P.1 Real Numbers and Algebraic Expressions
A ______is a collection of objects whose contents can be clearly determined. The objects in a set are called ______of the set. They are written in ______{ }.
Real Numbers
Rational Numbers Irrational Numbers
Integers
Whole Numbers
Natural Numbers
The set of ______is formed by combining the rational numbers and irrational numbers. The ______is a graph used to represent the set of real numbers.
Definition of Absolute Value
Example 1: Rewrite each expression without absolute value bars:
a) b) c) if x < 0 d)
Properties of Absolute Value
For all real numbers a and b,
1) 2) 3) 4)
5) b0 6) (called the triangle inequality)
Distance between Two Points on the Real Number Line
If a and b are any two points on a real number line, then the distance between a and b is given by or
Example 2: Find the distance between -5 and 3 on the real number line.
Algebra uses letters, such as x and y, to represent real numbers called ______. A combination of variables and numbers using the operations of addition, subtraction, multiplication, and division, as well as powers or roots, is called an ______.
Order of Operations (in order from left to right)
1) Parentheses
2) Exponents (powers)
3) Multiplication and division
4) Addition and subtraction
Example 3: Evaluate the following when x = 9
Name / Meaning / ExamplesCommutative property of addition/multiplication / a + b = b + a
ab = ba
Associative property of addition/multiplication / (a+b) + c = a + (b+c)
(ab)c = a(bc)
Distributive Property of multiplication over addition / a(b + c) = ab + ac
Identity property of addition/multiplication / a + 0 = a ; 0 + a = a
a(1) = a ; 1(a) = a
Inverse property of addition/multiplication / a + (-a) = 0; (-a) + a = 0
; a0
We call the ______or ______of b. The quotient of a and b, , can be written in the form where a is the ______and b is the ______of the fraction.
Simplifying Algebraic Expressions
5x – 6y + 2
Example 4: Simplify 6(2x – 4y) + 10(4x + 3y)
Properties of Negatives:
1) (-1)a = -a 2) -(-a) = a 3) (-a)b = -ab 4) a(-b) = -ab
5) -(a + b) = -a – b 6) -(a – b) = -a + b